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arXiv:2601.04407

Exact Multimode Quantization of Superconducting Circuits via Boundary Admittance and Continued Fractions

Mustafa Bakr, Robin Wopalenski

quant-phcond-mat.mes-hallmath-phSubmitted Jan 7, 2026• Announced Jan 9, 2026• 0 citations• Revised Feb 4, 2026 (v3)
v1 · Jan 7, 2026v2 · Jan 11, 2026v3 · Feb 4, 2026
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Abstract

Accurate extraction of linearized quantum circuit models from electromagnetic simulations is essential for the design of superconducting circuits. We present a quantization framework based on the driving-point admittance $Y_{\mathrm{in}}(s)$ seen by a Josephson junction embedded in an arbitrary passive linear environment. By taking the Schur complement of the nodal admittance matrix, we show that the linearized coupled system obeys an eigenvalue-dependent boundary condition, $s Y_{\mathrm{in}}(s) + 1/L_J = 0$, whose roots determine the dressed linear mode frequencies. This boundary condition admits an exact continued fraction representation: any positive-real admittance can be realized as a canonical Cauer ladder, yielding a tridiagonal (Jacobi) structure that enables certified convergence bounds via interlacing theorems.For the full nonlinear Hamiltonian, we treat Josephson junctions in the charge basis, where each cosine potential is exactly tridiagonal, and couple them to cavity modes in the Fock basis; in the general multi-junction case this yields a block-tridiagonal structure solvable by matrix continued fractions, enabling systematic diagonalization across all coupling regimes from dispersive through ultrastrong and deep strong coupling. The resulting quantization procedure is: (i)~compute or measure $Y_{\mathrm{in}}(s)$, (ii)~solve the boundary condition to obtain dressed eigenfrequencies, (iii)~synthesize an equivalent passive network, and (iv)~quantize while retaining the full cosine nonlinearity of the Josephson junction. We prove that junction participation decays as $\mathcal{O}(ω_n^{-1})$ at high frequencies for any circuit with finite shunt capacitance, ensuring ultraviolet convergence of perturbative corrections without imposed cutoffs.

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•Sep 1, 2019•0 citations
Accurate extraction of linearized quantum circuit models from electromagnetic simulations is essential for the design of superconducting circuits. We present a quantization framework based on the driving-point admittance Yin(s)Y_{\mathrm{in}}(s)Yin​(s) seen by a Josephson junction embedded in an arbitrary passive linear environment. By taking the Schur complement of the nodal admittance matrix, we show that the linearized coupled system obeys an eigenvalue-dependent boundary condition, sYin(s)+1/LJ=0s Y_{\mathrm{in}}(s) + 1/L_J = 0sYin​(s)+1/LJ​=0, whose roots determine the dressed linear mode frequencies. This boundary condition admits an exact continued fraction representation: any positive-real admittance can be realized as a canonical Cauer ladder, yielding a tridiagonal (Jacobi) structure that enables certified convergence bounds via interlacing theorems.For the full nonlinear Hamiltonian, we treat Josephson junctions in the charge basis, where each cosine potential is exactly tridiagonal, and couple them to cavity modes in the Fock basis; in the general multi-junction case this yields a block-tridiagonal structure solvable by matrix continued fractions, enabling systematic diagonalization across all coupling regimes from dispersive through ultrastrong and deep strong coupling. The resulting quantization procedure is: (i)~compute or measure Yin(s)Y_{\mathrm{in}}(s)Yin​(s), (ii)~solve the boundary condition to obtain dressed eigenfrequencies, (iii)~synthesize an equivalent passive network, and (iv)~quantize while retaining the full cosine nonlinearity of the Josephson junction. We prove that junction participation decays as O(ωn−1)\mathcal{O}(ω_n^{-1})O(ωn−1​) at high frequencies for any circuit with finite shunt capacitance, ensuring ultraviolet convergence of perturbative corrections without imposed cutoffs.
Building on the first-order circuit quantization method [arXiv:2304.12252, arXiv:2401.09120], we provide simple formulas to construct exact Hamiltonians for Josephson-junction-based superconducting qudits capacitively, inductively, or galvanically coupled to passive linear environments. These environments may be multiport, multimode, discrete or continuous, reciprocal or nonreciprocal, and are characterized directly by their impedance or admittance matrices. In the weak-coupling regime, we further derive \emph{divergence-free} dispersive Hamiltonians for mode-resolved environments and transition-resolved weak-coupling master equations for dissipative continua. Mode structure, frequency renormalizations, environment-mediated interactions, decay rates, and directional cross couplings then follow from the same causal immittance response, while spurious Lamb-shift divergences arising from uncontrolled approximations in previous treatments are made explicit and avoided. We apply the theory to a set of illustrative circuits comprising a discrete resonator filter, finite-band metamaterial environments, nonreciprocal waveguide-QED systems, and superconducting giant atoms, for which analytical response matrices can be obtained, although the method is particularly well suited to numerical responses from electromagnetic solvers or experimental characterization. We thereby extend the black-box quantization framework to multiport, dissipative, and nonreciprocal settings, establishing a simple and scalable route toward optimized and automated electromagnetic design of large-scale superconducting quantum hardware.
In this work, we introduce new methods for the quantization, decomposition, and extraction (from electromagnetic simulations) of lumped-element circuit models for superconducting quantum devices. Our flux-charge symmetric procedures center on the network matrix, which encodes the connectivity of a circuit's inductive loops and capacitive nodes. First, we use the network matrix to demonstrate a simple algorithm for circuit quantization, giving novel predictions for the Hamiltonians of circuits with both Josephson junctions and quantum phase slip wires. We then show that by performing pivoting operations on the network matrix, we can decompose a superconducting circuit model into its simplest equivalent "fundamental" form, in which the harmonic degrees of freedom are separated out from the Josephson junctions and phase slip wires. Finally, we illustrate how to extract an exact, transformerless circuit model from electromagnetic simulations of a device's hybrid admittance/impedance response matrix, by matching the lumped circuit's network matrix to the network topology of the physical layout. Overall, we provide a toolkit of intuitive methods that can be used to construct, analyze, and manipulate superconducting circuit models.
We apply the momentous quantum mechanics formalism to the non-perturbative study of superconducting circuits in the transmon regime, deriving effective equations of motion for the bare Josephson junction (JJ), the cavity--JJ system, and a dissipative resonator. A Gaussian closure on the hierarchy of quantum moments resums the full cosine nonlinearity into the closed-form effective Hamiltonian 12[V(φ+φs)+V(φ−φs)]\tfrac{1}{2}[V(φ+φ_s)+V(φ-φ_s)]21​[V(φ+φs​)+V(φ−φs​)], which is non-perturbative and preserves the periodicity and boundedness of the Josephson potential at all phase amplitudes; the standard Kerr (Duffing) approximation is recovered as a special case. We derive the quantum-dressed frequency ωeff=Ωpcos⁡(φzpf/φ0)ω_{\rm eff}=Ω_p\sqrt{\cos(φ_{\rm zpf}/φ_0)}ωeff​=Ωp​cos(φzpf​/φ0​)​, and benchmark the effective dynamics against exact Mathieu-function diagonalization, with the quantum width G2,0(t)G^{2,0}(t)G2,0(t) providing the most sensitive diagnostic of the closure's validity and marks the boundary of the Gaussian approximation more sharply than ⟨φ^⟩(t)\langle\hatφ\rangle(t)⟨φ^​⟩(t) does. We compare the Caldirola--Kanai, Bateman, and Lindblad descriptions: the Bateman dynamics, quantized with a switched symplectic structure, preserves the Heisenberg bound, admits exact closed-form moment solutions, and reproduces the Lindblad benchmark for weak damping, i.e., within the analytic error bound (λ/ω1)2 φzpf2(λ/ω_1)^2\,φ_{\rm zpf}^2(λ/ω1​)2φzpf2​.
Josephson junctions are the central nonlinear element of superconducting qubits, yet their standard description is reached through a chain of approximations. We propose a purely quantum-mechanical model of the junction, formulated directly in Hilbert space: a charged particle moves on a one-dimensional loop, and the junction is a point at which the wave function obeys self-adjoint boundary conditions, labelled by a coupling strength λλλ, which controls the transparency of the junction, and a phase jump δδδ, which plays the role of the superconducting phase difference. For a loop with a single junction we solve the spectral problem and show that the ground-state momentum expectation, identified with the supercurrent, factorises exactly as ⟨p^0⟩=Ic(δ,λ)sin⁡δ\langle\hat{p}_0\rangle = I_c(δ,λ)\sinδ⟨p^​0​⟩=Ic​(δ,λ)sinδ. For opaque junctions IcI_cIc​ becomes independent of δδδ and the first Josephson relation I=Icsin⁡δI=I_c\sinδI=Ic​sinδ is recovered; for strongly attractive coupling, where the ground state is bound to the junction, the sin⁡δ\sinδsinδ law is also recovered, but with an exponentially suppressed amplitude. For more transparent junctions, however, the model \emph{predicts} deviations from the pure sin⁡δ\sinδsinδ law: IcI_cIc​ acquires a dependence on δδδ already at order λ−2λ^{-2}λ−2, compatible with the higher Josephson harmonics recently found necessary to describe superconducting qubit spectra, and a fully transparent contact yields a current linear in δδδ. When the phase varies in time, a gauge transformation shows that the voltage drop across the junction is proportional to δ˙\dotδδ˙, so that the second Josephson relation emerges without being imposed. The boundary-condition approach thus reproduces both Josephson relations from first principles and predicts corrections to them, offering a route towards higher-fidelity models of superconducting qubits.
The superconductor-insulator-superconductor Josephson junction is the fundamental nonlinear element of superconducting circuits. Connecting two junctions in series gives rise to higher-harmonic content in the total energy-phase relation, enabling new design opportunities in multimode circuits. However, the double-junction element hosts an internal mode whose spectrum is set by the finite capacitances of the individual junctions. Using a Born-Oppenheimer approximation that treats the additional mode as fast compared to the qubit mode, we analyze the double-junction circuit element shunted by a large capacitor. Here, we derive an effective single-mode model of the qubit containing a correction term owing to the presence of the internal mode. In experimentally relevant parameter regimes, we numerically find that our model accurately describes the low-energy spectrum of the qubit. We further discuss how eliminating the internal degree of freedom affects the system's periodic boundary conditions and leads to non-uniqueness in performing the Born-Oppenheimer approximation. Finally, we analyze the harmonic content of the double-junction element and discuss its sensitivity to charge noise.
We present a semi-classical method for determining the effective low-energy quantum Hamiltonian of weakly anharmonic superconducting circuits containing mesoscopic Josephson junctions coupled to electromagnetic environments made of an arbitrary combination of distributed and lumped elements. A convenient basis, capturing the multi-mode physics, is given by the quantized eigenmodes of the linearized circuit and is fully determined by a classical linear response function. The method is used to calculate numerically the low-energy spectrum of a 3D-transmon system, and quantitative agreement with measurements is found.